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METHOD 1:

Let's divide some regular polygons into triangles by connecting one vertex to all of the others...

sqr2.gif

 

pent.gif

 

hx.gif

A square has 4 sides and we made 2 triangles.

    

A pentagon has 5 sides and we made 3 triangles.

    

A hexagon has 6 sides and we made 4 triangles.

Do you see the pattern?

A heptagon has 7 sides... so we'd be able to make 5 triangles.

If we had polygon with n sides... we'd be able to make (n - 2) triangles.

Let's start with the square... We made 2 triangles. Notice that all of the interior angles of the 2 triangles make up the interior angles of the square.

The sum of the 2 triangle's angles is  degrees0.gif

There are 4 equal angles in a square,

so degrees0a.gif gives us that one angle of a square is degrees0b.gif !

Just what we expected.

Now for the pentagon.

pent.gif

We made 3 triangles.

degrees2.gif

So the interior angles of a regular pentagon are each 108 degrees.

Can you figure out the hexagon?

hx.gifdegrees3.gif

How about a 100-gon? (That's a regular polygon with 100 sides.) There would be 98 triangles...

degrees4.gif

So, in general, the measure of an interior angle of a regular n-gon is

degrees5.gif